Problem: Simplify and expand the following expression: $ \dfrac{x}{x - 4}+\dfrac{2x}{3x + 5} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(x - 4)(3x + 5)$ Multiply the first term by $\dfrac{3x + 5}{3x + 5}$ $ \begin{align*} \dfrac{x}{x - 4} \times \dfrac{3x + 5}{3x + 5} & = \dfrac{(x)(3x + 5)}{(x - 4)(3x + 5)} \\ & = \dfrac{3x^2 + 5x}{(x - 4)(3x + 5)}\end{align*} $ Multiply the second term by $\dfrac{x - 4}{x - 4}$ $ \begin{align*} \dfrac{2x}{3x + 5} \times \dfrac{x - 4}{x - 4} & = \dfrac{(2x)(x - 4)}{(3x + 5)(x - 4)} \\ & = \dfrac{2x^2 - 8x}{(3x + 5)(x - 4)}\end{align*} $ Now we have: $ = \dfrac{3x^2 + 5x}{(x - 4)(3x + 5)} + \dfrac{2x^2 - 8x}{(3x + 5)(x - 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3x^2 + 5x + 2x^2 - 8x}{(x - 4)(3x + 5)} $ $ = \dfrac{5x^2 - 3x}{(x - 4)(3x + 5)}$ Expand the denominator: $ = \dfrac{5x^2 - 3x}{3x^2 - 7x - 20}$